Describe the graph of each equation as a circle, a point, or nonexistent. If it is a circle, give the center and radius. If it is a point, give the coordinates. x²+y²-2x+12y-12=0

Verified step by step guidance

1. Start by rewriting the given equation: \(x^2 + y^2 - 2x + 12y - 12 = 0\).
2. Group the \(x\) terms and \(y\) terms together: \((x^2 - 2x) + (y^2 + 12y) = 12\) (move the constant to the right side).
3. Complete the square for the \(x\) terms: take half of \(-2\), which is \(-1\), and square it to get \$1\(. Add \)1\( inside the \)x$ group.
4. Complete the square for the \(y\) terms: take half of \$12\(, which is \)6\(, and square it to get \)36\(. Add \)36\( inside the \)y$ group.
5. Since you added \$1\( and \)36\( to the left side, add \)1 + 36 = 37\( to the right side as well to keep the equation balanced. Then rewrite the equation as \)(x - 1)^2 + (y + 6)^2 = 12 + 37$.

Key Concepts

Standard Form of a Circle Equation

The standard form of a circle's equation is (x - h)^2 + (y - k)^2 = r^2, where (h, k) is the center and r is the radius. Converting a general quadratic equation into this form helps identify the circle's properties.

Completing the Square

Completing the square is a method used to rewrite quadratic expressions as perfect square trinomials. This technique is essential to transform the given equation into the standard form of a circle.

Determining the Nature of the Graph

After rewriting the equation, the value of r^2 determines the graph's nature: if r^2 > 0, it's a circle; if r^2 = 0, it's a single point; if r^2 < 0, the graph does not exist in the real plane.